TITLE: Mistake in Popescu's book "Abelian Categories with Applications to Rings and Modules"
QUESTION [5 upvotes]: Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says:

Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and faithful right adjoint of $F$. Then:
  If $C$ is a category with inductive (respectively, projective) limits, then $C^\prime$ is also a category with inductive (respectively, projective) limits.

The statement about the existence of colimits (inductive limits) is clear, but I think his "respectively" statement is wrong.
This could be just a typo if they forgot to include something like "[...] and $G$ be a full and faithful right (respectively left) adjoint of $F$[...]", but I think he uses the wrong version of this corollary later:
On page 111 in chapter 3 he states and proves the famous Gabriel-Propescu theorem (Theorem 7.9):

Let $C$ be a Grothendieck category, $U$ an generator of $C$ and $A=Hom_C(U,U)$. Also, let $T=\_\otimes_A U\colon Mod(A^{op})\rightarrow C$ and $S$ its right adjoint. Then $S$ is full and faithful and $T$ is exact.

In the next corollary (7.10) after the proof of the theorem, he states that any Grothendieck category is complete (has (projective) limits) and just cites corollary 5.5 a) as a proof.
My question is: Did he really made a mistake in this proof? What would be an argument to follow the completeness of a Grothendieck category from the Gabriel-Popescu theorem instead? (For sure one can show the completeness without Gabriel-Popescu, for example like it is done in Kashiwara & Schapira's Categories and Sheaves.)

REPLY [5 votes]: Despite the formulation of the claim, there are really two results in there which have different proofs.

If $F : \mathcal{C} \to \mathcal{C}'$ is a functor with a fully faithful right adjoint $G : \mathcal{C}' \to \mathcal{C}$ and $\mathcal{C}$ is cocomplete, then $\mathcal{C}'$ is also cocomplete: given any diagram $X : \mathcal{J} \to \mathcal{C}'$, we have 
$$F (\varinjlim_\mathcal{J} G X) \cong \varinjlim_\mathcal{J} F G X \cong \varinjlim_\mathcal{J} X$$
and so $X$ indeed has a colimit in $\mathcal{C}'$.
If $F : \mathcal{C} \to \mathcal{C}'$ is a functor with a fully faithful right adjoint $G : \mathcal{C}' \to \mathcal{C}$ and $\mathcal{C}$ is complete, then $\mathcal{C}'$ is also complete: this because $G F$ is (the underlying endofunctor of) an idempotent monad on $\mathcal{C}$ and $\mathcal{C}'$ is (isomorphic to) the Eilenberg–Moore category for $G F$, and it is well-known that the Eilenberg–Moore category of a monad is complete if the base category is complete.